Triangular Factorization and Inversion by Fast Matrix Multiplication

o to N Q < TRIANGULAR FACTORIZATION AND INVERSION BY FAST MATRIX MULTIPLICATION

High-performance and Parallel Inversion of a Symmetric Positive Definite Matrix

We present families of algorithms for operations related to the computation of the inverse of a Symmetric Positive Definite (SPD) matrix: Cholesky factorization, inversion of a triangular matrix, multiplication of a triangular matrix by its transpose, and one-sweep inversion of an SPD matrix. These algorithms are systematically derived and implemented via the Formal Linear Algebra Methodology E...

Stability of Methods for Matrix Inversion

Inversion of a triangular matrix can be accomplished in several ways. The standard methods are characterised by the loop ordering, whether matrix-vector multiplication, solution of a triangular system, or a rank-1 update is done inside the outer loop, and whether the method is blocked or unblocked. The numerical stability properties of these methods are investigated. It is shown that unblocked ...

Incomplete Cholesky Parallel Preconditioners with Selective Inversion

Consider the solution of a large linear system when the coe cient matrix is sparse, symmetric, and positive de nite. One approach is the method of \conjugate gradient" (CG) with an incomplete Cholesky (IC) preconditioner (ICCG). A key problem with the design of a parallel ICCG solver is the bottleneck posed by repeated parallel sparse triangular solves to apply the preconditioner. Our work conc...

Contemporary Mathematics The Schur algorithm for matrices with Hessenberg displacement structure

A Schur-type algorithm is presented for computing recursively the triangular factorization R = LU of a strongly nonsingular n n matrix satisfying a displacement equation RY V R = GH with Hessenberg matrices Y and V and n matrices G, H. If is small compared with n and the matrices Y and V admit fast matrix-vector multiplication, the new algorithm is fast in the sense that it will require less th...